R: Vertex connectivity.

vertex_connectivity {igraph} | R Documentation

## Vertex connectivity.

### Description

The vertex connectivity of a graph or two vertices, this is recently also called group cohesion.

### Usage

    vertex_connectivity(graph, source = NULL, target = NULL,
      checks = TRUE)

    ## S3 method for class 'igraph'
    cohesion(x, checks = TRUE, ...)

### Arguments

`graph, x` | The input graph.
`source` | The id of the source vertex, for `vertex_connectivity` it can be `NULL`, see details below.
`target` | The id of the target vertex, for `vertex_connectivity` it can be `NULL`, see details below.
`checks` | Logical constant. Whether to check that the graph is connected and also the degree of the vertices. If the graph is not (strongly) connected then the connectivity is obviously zero. Otherwise if the minimum degree is one then the vertex connectivity is also one. It is a good idea to perform these checks, as they can be done quickly compared to the connectivity calculation itself. They were suggested by Peter McMahan, thanks Peter.
`...` | Ignored.

### Details

The vertex connectivity of two vertices (`source` and `target`) in a directed graph is the minimum number of vertices needed to remove from the graph to eliminate all (directed) paths from `source` to `target`. `vertex_connectivity` calculates this quantity if both the `source` and `target` arguments are given and they're not `NULL`.

The vertex connectivity of a graph is the minimum vertex connectivity of all (ordered) pairs of vertices in the graph. In other words this is the minimum number of vertices needed to remove to make the graph not strongly connected. (If the graph is not strongly connected then this is zero.) `vertex_connectivity` calculates this quantitty if neither the `source` nor `target` arguments are given. (Ie. they are both `NULL`.)

A set of vertex disjoint directed paths from `source` to `vertex` is a set of directed paths between them whose vertices do not contain common vertices (apart from `source` and `target`). The maximum number of vertex disjoint paths between two vertices is the same as their vertex connectivity in most cases (if the two vertices are not connected by an edge).

The cohesion of a graph (as defined by White and Harary, see references), is the vertex connectivity of the graph. This is calculated by `cohesion`.

These three functions essentially calculate the same measure(s), more precisely `vertex_connectivity` is the most general, the other two are included only for the ease of using more descriptive function names.

### Value

A scalar real value.

### Author(s)

Gabor Csardi csardi.gabor@gmail.com

### References

White, Douglas R and Frank Harary 2001. The Cohesiveness of Blocks In Social Networks: Node Connectivity and Conditional Density. Sociological Methodology 31 (1) : 305-359.

### See Also

`max_flow`, `edge_connectivity`, `edge_disjoint_paths`, `adhesion`

### Examples

    g <- barabasi.game(100, m=1)
    g <- delete_edges(g, E(g)[ 100 %--% 1 ])
    g2 <- barabasi.game(100, m=5)
    g2 <- delete_edges(g2, E(g2)[ 100 %--% 1])
    vertex_connectivity(g, 100, 1)
    vertex_connectivity(g2, 100, 1)
    vertex_disjoint_paths(g2, 100, 1)

    g <- sample_gnp(50, 5/50)
    g <- as.directed(g)
    g <- induced_subgraph(g, subcomponent(g, 1))
    cohesion(g)

[Package igraph version 1.2.4.1 Index]